On divided differences on C[a,b], \(L_ p[a,b]\) and other spaces
The rth difference is defined by \(\Delta_ hf(x):=f(x+h)-f(x)\) and \(\Delta^ r_ hf(x) := \Delta_ h(\Delta_ h^{r-i}f(x))\). If \(h_ n=O(1)(n\to \infty)\) and \(1\leq h_ n/h_{n+1}\leq M\), then \(\| \Delta^ r_{h_ n}f\|_ B\leq Mh_ n^{\alpha}\Rightarrow \| \Delta^ r_ hf\|_ B\leq Kh^{\alpha}\) for different spaces B (R. De Vore, G. Freud, the author, J. Boman, V. Totik). More generally, for \(0=h_ 0(n)<h_ 1(n)<...<h_ r(n)\), divided difference is defined by \[ [0,h_ 1(n),...,h_ r(n)]f(x+\cdot):=(1)^ r\sum^{r}_{i=0}f(x+h_ i(n))/W'(h_ i(n)) \] where \(W(t)=\prod^{r}_{i=0}(t-h_ i(n))\) (in particular, \([0,h,2h,...,rh]f(x+\cdot)=(h^ rr!)^{-1}\Delta^ r_ hf(x))\). The author proves that for many Banach spaces B: (i) If the sequence of vectors \((h_ 0(n),...,h_ r(n))\) satisfies \(h_ i(n)-h_{i-1}(n)\geq Ah_ r(n),\) \(h_ r(n)=O(1)\) (n\(\to \infty)\) and \(1\leq h_ r(n)/h_ r(n+1)\leq M\), then \(\| [0,h_ 1(n),...,h_ r(n)]f(x+\cdot)\|_ B\leq Mh_ r(n)^{-r+\alpha}\) is equivalent to \(\| \Delta^ r_ hf\|_ B\leq Kh^{\alpha}\). (ii) If \(Ah_ 1(n)\leq h_ i(n)\leq Bh_ 1(n)\), \(h_ 1(n)=O(1)\) (n\(\to \infty)\) and \(1\leq h_ 1(n)/h_ 1(n+1)\leq M\) then \[ \| \Delta_{h_ 1(n)}...\Delta_{h_ r(n)}f(x)\|_ B\leq Kh_ 1(n)^{\alpha}\Leftrightarrow \| \Delta^ r_ hf\|_ B\leq Kh^{\alpha}. \]
scientific article; zbMATH DE number 4009008
scientific article